Geodesic
Hilbert symbol in a discrete valuated field
Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 50-69 
S. V. Vostokov. Hilbert symbol in a discrete valuated field. Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 50-69. http://geodesic.mathdoc.fr/item/ZNSL_1979_94_a5/
@article{ZNSL_1979_94_a5,
     author = {S. V. Vostokov},
     title = {Hilbert symbol in a discrete valuated field},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {50--69},
     year = {1979},
     volume = {94},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_94_a5/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In a complete discrete valuated field of characteristic zero with perfect residue field of prime characteristic a pairing with values in the group of primary elements is defined. It is proved that this pairing agrees with the Hilbert norm residue symbol and therefore provides an explicit formula for the latter.